Original citation:
Pugh, C. E., Nakariakov, V. M. and Broomhall, Anne-Marie. (2015) A multi-period oscillation
in a stellar superflare. Astrophysical Journal Letters, 813 (1). L5.
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The Astrophysical Journal Letters, 813:L5 (5pp), 2015 November 1
doi:10.1088/2041-8205/813/1/L5
© 2015. The American Astronomical Society. All rights reserved.
A MULTI-PERIOD OSCILLATION IN A STELLAR SUPERFLARE
C. E. Pugh1, V. M. Nakariakov1,2, and A.-M. Broomhall1,3
1
Centre for Fusion, Space, and Astrophysics, Department of Physics, University of Warwick, Coventry CV4 7AL, UK; [email protected]
2
Central Astronomical Observatory at Pulkovo of RAS, St Petersburg 196140, Russia
3
Institute of Advanced Study, University of Warwick, Coventry CV4 7HS, UK
Received 2015 May 15; accepted 2015 October 9; published 2015 October 23
ABSTRACT
Flares that are orders of magnitude larger than the most energetic solar flares are routinely observed on Sun-like
stars, raising the question of whether the same physical processes are responsible for both solar and stellar flares. In
this Letter, we present a white-light stellar superflare on the star KIC 9655129, observed by NASA’s Kepler
mission, with a rare multi-period quasi-periodic pulsation (QPP) pattern. Two significant periodic processes were
detected using the wavelet and autocorrelation techniques, with periods of 78 ± 12 minutes and 32 ± 2 minutes.
By comparing the phases and decay times of the two periodicities, the QPP signal was found to most likely be
linear, suggesting that the two periodicities are independent, possibly corresponding either to different
magnetohydrodynamic (MHD) modes of the flaring region or different spatial harmonics of the same mode.
The presence of multiple periodicities is a good indication that the QPPs were caused by MHD oscillations and
suggests that the physical processes in operation during stellar flares could be the same as those in solar flares.
Key words: stars: activity – stars: coronae – stars: flare – stars: oscillations – Sun: flares – Sun: oscillations
1. INTRODUCTION
modulation of particle acceleration and the rate of energy
release. Examples of this regime have been found in numerical
simulations by Murray et al. (2009), Kliem et al. (2000), and
McLaughlin et al. (2012). QPPs caused by MHD oscillations
could involve either oscillations of the flaring region itself or
MHD oscillations of a nearby structure. In the first case,
variation of parameters of the flaring plasma (such as the
magnetic field and plasma density) could directly modulate the
radiation emission or could result in periodic modulation of
particle acceleration and hence emission via the gyrosynchrotron mechanism or bremsstrahlung. The latter case may be
considered a periodically triggered regime of magnetic
reconnection, where the fast or slow magnetoacoustic oscillations leak from the oscillating structure and approach the
flaring site, resulting in, for example, plasma microinstabilities
and hence anomalous resistivity, which could trigger reconnection in the flaring loop (Chen & Priest 2006; Nakariakov
et al. 2006).
The first observation of oscillations in a stellar flare was
made by Rodono (1974). Since then, occasional observations
of QPPs in different stars have been made in the optical
(Mathioudakis et al. 2003), ultraviolet (Welsh et al. 2006),
microwave (Zaitsev et al. 2004), and X-ray (Mitra-Kraev
et al. 2005) wavebands. Recently, Anfinogentov et al. (2013)
found QPPs in a megaflare on the dM4.5e star YZ CMi,
observed in white light, which looked very similar to
oscillations in solar flares that were concluded to be the result
of standing longitudinal modes. This suggests that this
mechanism, where the oscillations cause plasma parameters
to vary and hence modulate the acceleration of precipitating
non-thermal electrons, applies to a wide range of flare energies,
including superflares. So far, no other evidence has been found
to suggest any major differences between solar and stellar
QPPs, indicating that the basic physical processes responsible
for the energy releases (e.g., magnetic reconnection) are
the same.
NASA’s Kepler mission is proving to be an excellent
resource for the study of stellar flares, owing to the large
Solar flares typically release 10 –10 erg of energy over a
timescale of up to several hours. Many stars not dissimilar from
the Sun have been observed to produce flares several orders of
magnitude more powerful than any solar flare on record, with
amplitudes around 0.1%–1% of the stellar luminosity and
estimated energies of 1033–1036 erg (Maehara et al. 2012). Due
to the potential for substantial damage in the near-Earth
environment associated with powerful flares, it is important to
work toward determining whether a superflare could occur on
the Sun, and if so, what the probability of one occurring at a
given time would be. Shibata et al. (2013) found that, in
principle, a 1034 erg superflare could occur on the Sun if the
necessary magnetic flux were stored over one solar cycle period
(∼11 years), while it would take around 40 years to accumulate
enough magnetic flux for a 1035 erg superflare. It is not
completely understood how magnetic flux could be prevented
from emerging from the base of the convection zone for such a
long period of time, hence additional observational data and
theoretical constraints are needed to support or disprove these
findings.
Quasi-periodic pulsations (QPPs) are time variations in the
intensity of light emitted by a flare. They have been widely
observed in solar flares (e.g., Kupriyanova et al. 2010), and
since they are a common feature of flares, QPPs can be used to
constrain parameters of the plasma in the flaring region and
indicate the physical processes in operation. The specific
mechanism responsible for QPPs is unknown, but several
possible mechanisms have been proposed. These fall into one
of two categories: “load/unload” (or “magnetic dripping”)
mechanisms or magnetohydrodynamic (MHD) oscillations
(e.g., Nakariakov & Melnikov 2009; Nakariakov et al. 2010).
The load/unload mechanisms could apply where there is a
continuous supply of magnetic energy, causing magnetic
reconnection to repetitively occur each time a threshold energy
is surpassed. In this scenario, the periodicity is connected with
a quasi-periodic self-oscillatory regime of spontaneous magnetic reconnection, which would result in quasi-periodic
29
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The Astrophysical Journal Letters, 813:L5 (5pp), 2015 November 1
Pugh, Nakariakov, & Broomhall
candidate flaring A- to M-type stars, and these were then
checked by eye. Any flares showing potential signs of QPPs
were analyzed using the wavelet and autocorrelation techniques
(detailed below). Of the 2982 flares detected on 215 stars using
this method, 73 showed evidence of QPPs, 9 of which had
stable periodicities, and 1 was found with multiple significant
periodicities. This is the flare studied in the remainder of this
paper. A section of the light curve from KIC 9655129
containing the flare of interest is shown in the left-hand panel
of Figure 1. The periodic modulation of the light curve is due to
this star being a binary, and the dip is where one star eclipses
the other. The three small peaks near the center are flares, and
the right-hand panel of Figure 1 shows the decaying phase of
the central flare.
To help detect QPPs, the decaying phase of the flare light
curve was fitted with an aperiodic analytical function using a
least-squares method, and the fit was then subtracted from the
light curve. Details of the fitting are given in Section 3. The
resultant de-trended light curve was padded with zeros at the
start, and a wavelet transform was then performed using the
Morlet wavelet to highlight any periodicities. In order to obtain
estimates for the periods of the two detected periodicities, a
function (described in Section 3) was fitted to the flare decay,
and uncertainties were estimated using Monte Carlo
simulations.
Another useful technique for reducing noise and highlighting
periodicities in the data is the autocorrelation method, where
the correlation of a signal with itself is calculated as a function
of time lag. In this study, the time lags used ranged from zero to
half the duration of the flare decay time series, with a spacing
equal to the cadence of the data. The periodic variability of the
autocorrelation function was then studied.
number of stars observed over long periods of time, the high
sensitivity of its photometric observations, and the availability
of data with a cadence of one minute: suitable for studying
QPPs with periods greater than a few minutes. Balona et al.
(2015) found several instances of quasi-periodic variability in
stellar flares using Kepler data, the periods of which were not
found to correlate with any global stellar parameter, suggesting
that they could be QPPs in the impulsive energy release itself.
In this study, we focus on a rare example of a flare showing
evidence of two periods of oscillation, which occurred on KIC
9655129, a K-type eclipsing binary star. While it is difficult to
tell from which of the binary stars the flare originates at this
stage, the conclusions of this paper do not rely upon the stellar
spectral type. This star has a low contamination factor of 0.035,
meaning that little of the light detected comes from surrounding
objects. Multiple-period QPPs are of great interest because they
impose additional constraints on the plasma parameters, have
implications for the underlying QPP mechanisms, and have
interesting seismological implications (e.g., Van Doorsselaere
et al. 2007; Inglis & Nakariakov 2009; Van Doorsselaere
et al. 2011). Detections of multiple-period QPPs are only very
occasionally made in solar flares (e.g., Inglis & Nakariakov 2009), owing to the lower amplitudes of higher harmonics,
and they are even rarer in stellar flares, most likely due to the
noise level of the data. Only two examples in stellar flares have
been reported previously, both of which were detected in
different wavebands to the flare presented in this paper. The
first case was found in the microwave band by Zaitsev et al.
(2004), which had one quasi-periodic component with a period
that varied from 0.5 to 2 s during the flare, and the other was a
series of pulses with a period of 0.5 s. The second case was
found in the X-ray band by Srivastava et al. (2013), which had
periods of 21.0 and 11.5 minutes. For these two cases, the
detected periodicities are different from the ones found in our
study, and the flares themselves may or may not be superflares
since the flare amplitudes were not specified. Moreover, a
white-light emission burst is considered to be a signature of a
superflare. An overview of the data and analysis methods
employed to deduce the periods of oscillation are given in
Section 2, while the results and details of the analysis are given
in Section 3. A summary is given in Section 4.
3. RESULTS AND DISCUSSION
In order to remove the underlying trend in the flare decay
light curve, and hence emphasize any short-term variability, the
following expression was fitted to the light curve, as shown by
the red overplotted line in the right-hand plot of Figure 1:
F (t ) = Ae-t
t0
+ Bt + C,
(1 )
where F is the flux, t is time, and A, t0, B, C are constants. An
exponential decay expression was chosen because, for most
cases, it fits the decaying phase of the flares well. One
complication is underlying trends in the light curves, which are
usually due to differential velocity aberration, rotational
variability (if the star is a binary or has starspots), and/or
transits. The addition of a linear term, Bt, was enough to
account for the underlying trend in the vicinity of the flare for
this case.
To aid in visualizing the QPPs, an autocorrelation was
performed on the light curve after subtracting the decay trend,
as shown in Figure 2. The trend function in Equation (1) is
aperiodic, hence its subtraction from the signal cannot
introduce artificial periodicities. A decaying oscillation can
clearly be seen in this plot, and the following expression,
shown by the red overplotted line, fits very well:
2. DATA AND ANALYSIS
The majority of data from Kepler have a cadence of 30
minutes, but several thousand stars have also been observed
with a one-minute cadence, which is more appropriate for the
study of QPPs in flares. The flare of interest occurred on 2012
August 9 and can be found in the short-cadence light curve
from Quarter 14b. Two types of flux data are available for each
light curve: SAP and PDCSAP. The main difference is that
PDCSAP data have had artifacts and systematic trends
removed (Kinemuchi et al. 2012). The pipeline module used
to produce the PDCSAP flux is designed to optimize the data
for the detection of exoplanet transits; however, in some cases,
variations and artifacts that are astrophysical in nature are also
removed. Fortunately, SAP flux data are suitable for the study
of stellar flares, as the flares can easily be distinguished from
most artifacts by comparing with the PDCSAP data, and the
timescales of systematic trends are much greater than the
durations of the flares.
An algorithm similar to that detailed in Walkowicz et al.
(2011) was used to search all short-cadence Kepler data for
F (t ) = Ae-t
2
t0
⎛ 2p
⎞
cos ⎜ t + f⎟ + C ,
⎝ P
⎠
(2 )
The Astrophysical Journal Letters, 813:L5 (5pp), 2015 November 1
Pugh, Nakariakov, & Broomhall
Figure 1. Left: a section of the short-cadence light curve of KIC 9655129 from Quarter 14b, which contains three flares. Right: a shorter section of the light curve,
showing the decaying phase of the central flare in the plot on the left. The peak intensity of the flare is at time t = 0. The red overplotted line is the result of a leastsquares fit to the flare decay, as detailed in Section 3.
shows a second period of 32 ± 7 minutes. There is also
evidence of a possible third periodicity at around 19 minutes,
but it requires a more complex fit.
Figure 4 shows the result of fitting all parameters
simultaneously and performing 10,000 Monte Carlo simulations, using a fitting function combining Equation (1) with the
two periodicities, described by Equation (2). Histograms for the
two fitted periods are shown in Figure 5. These have been fitted
with Gaussians to give values of 78 ± 12 and 32 ± 2 minutes,
which are in good agreement with the values obtained using the
wavelet and autocorrelation methods. The same Monte Carlo
simulations were used to evaluate correlations between the
different fitted parameters. No strong correlations were found
between the periods themselves (8% Pearson’s correlation), or
between the periods and their corresponding decay times
(Pearson’s correlation of 19% for the longer periodicity and
29% for the shorter periodicity), suggesting that they are
independent. Although the longer periodicity was found to be
slightly correlated with the decay time of the flare itself (t0 in
Equation (1)), with a Pearson’s correlation of 50%, over 99%
+25
of the fitted values were within the range of 8419 minutes
indicated by the wavelet analysis, suggesting that the data detrending has not significantly influenced the results.
The significance of finding a multi-period QPP is that it is a
strong indication that MHD oscillations are the cause of the
QPPs in this flare. Multiple periods are difficult to explain
with the load/unload mechanisms, whereas harmonics are a
common feature of resonators, and different types of waves
have different characteristic periods and damping times. There
is, however, a possibility that the QPP signal detected in the
flare is nonlinear, and such a signal could readily be produced
by self-oscillation (Nakariakov et al. 2012). A nonlinear signal,
for example, a sawtooth wave, can be constructed by the
superposition of different sine/cosine waves (a Fourier series
expansion), hence its Fourier/wavelet spectrum would have
multiple peaks. In this case, the phase difference between the
sinusoidal components at a particular point in time would be 2π
less than the phase difference at a point one cycle of the
fundamental harmonic ahead of the first point. On the other
hand, if the signals belong to different MHD modes or their
spatial harmonics, they may have phases disconnected from
one another. To check whether the detected periodicities are
Figure 2. Autocorrelation function of the de-trended flare decay light curve
(shown in Figure 3(a)), with a fitted decaying sinusoid overplotted in red,
which has a period of 86.1 ± 5.4 minutes.
where P is the period, f is the phase, and A, t0, C are constants.
This fit gives a period of 86.1 ± 5.4 minutes, where the error
was obtained by performing Monte Carlo simulations.
Figure 3(a) shows the de-trended light curve. The wavelet
spectrum, shown in Figure 3(b), has a clear feature at a period
+25
of 8419 minutes, which is above the 99% confidence level (as
defined by Torrence & Compo 1998). There is also a feature
above the 99% level at a period of around 250 minutes, which
can be ignored as its duration is roughly equal to its period, so
it cannot be considered to be an oscillatory pattern. The small
short-period features are due to noise in the data. The highpower spectral feature appears to split into two bands,
suggesting the presence of two different periodicities.
To examine a possible second periodicity, we subtracted the
signal given by Equation (2) with the best-fitting coefficients
from the de-trended original light curve, and then performed a
wavelet transform on the resultant time series. The remnant
signal is shown in Figure 3(c), with a decaying sinusoidal fit
overplotted in red. Despite the noise in this plot, several cycles
of the oscillation can be seen near the start, and this becomes
more clear when a wavelet transform is performed, as shown in
Figure 3(d). The bright band above the 99% confidence level
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The Astrophysical Journal Letters, 813:L5 (5pp), 2015 November 1
Pugh, Nakariakov, & Broomhall
Figure 3. Time series are shown on the left with their corresponding wavelet spectra on the right. The far-right panels show the global wavelet spectra. The beginnings
of the time series used to produce panels (b) and (d) have been padded with zeros in order to bring the features of interest into the center of the cone of influence. In
each case, the peak of the flare is at the time t = 0. (a) The de-trended decay phase of the flaring light curve, with a fit to the main periodicity overplotted in red. (b) The
+25
wavelet spectrum of panel (a). The bright feature has a period of 8419 minutes. (c) The de-trended light curve in panel (a) with a fit to the main periodicity subtracted.
Fitting the shorter periodicity gives the curve overplotted in red. (d) The wavelet spectrum of panel (e), showing a second period of 32 ± 7 minutes.
longer periodicity subtracted (shown in Figure 3(c)). The
phases were found to be 3.8 ± 0.1 radians for the longer
periodicity and 0.6 ± 0.2 radians for the shorter periodicity,
giving a phase difference of 3.2 ± 0.2 radians at the time t = 0.
After one cycle of the longer periodicity, the phase difference is
5.7 ± 0.2 radians, and after two cycles, it is 14.6 ± 0.2 radians.
Since these phase differences do not differ by a factor of 2π,
this suggests that the signal is linear and that the shorter
periodicity is a spatial harmonic of the longer periodicity or the
result of a different MHD mode. Also, the decay time of the
shorter periodicity is 77 ± 29 minutes, compared to 80 ±
12 minutes for the longer periodicity. If this shorter periodicity
were a higher harmonic of a nonlinear signal, it would be
expected to decay faster than the fundamental harmonic.
Indeed, it is possible that the shorter periodicity decays faster
than the main oscillation due to the associated uncertainties, but
it is more likely the case that they are similar in duration.
Considering the ratio of the periods, for a uniform medium,
the fundamental and second harmonics might be expected to be
a factor of two different, whereas here the periods have a ratio
of 2.4 ± 0.4. While some MHD modes are dispersionless or
weakly dispersive (e.g., torsional and longitudinal), others are
highly dispersive (e.g., kink and sausage) and so the ratio of the
periods of their spatial harmonics can be non-integer (Inglis &
Nakariakov 2009). Also stratification of the plasma density due
Figure 4. Flare decay light curve with the red overplotted curve showing the
result of a least-squares fit to the flare decay, along with the two periodicities.
time harmonics of the same nonlinear signal, the phases of
oscillation of the two periodicities were found by fitting
Equation (2), with a period equal to 78 minutes, to the detrended light curve (shown in Figure 3(a)) and fitting the
shorter periodicity to the de-trended light curve with the fitted
4
The Astrophysical Journal Letters, 813:L5 (5pp), 2015 November 1
Pugh, Nakariakov, & Broomhall
Figure 5. Histograms showing the results of Monte Carlo simulations for the main (left) and secondary (right) periods. The red overplotted curves show Gaussian fits,
which have been used to estimate values of 78 ± 12 minutes for the longer period and 32 ± 2 minutes for the shorter period.
to gravity, along with the geometry of coronal loops, means
that the plasma density and magnetic field strength vary along
the loop, and hence it is most likely that the wave frequency
does not scale linearly with the wavelength for spatial
harmonics. Therefore, this ratio is consistent with the findings
in the previous paragraph, and the presence of spatial
harmonics. While we cannot use this information to conclusively identify the mechanism behind the QPPs, sausage modes
can be excluded, as their characteristic periods are much shorter
than those detected here (Nakariakov et al. 2012).
the data used in this paper, obtained from the Mikulski Archive
for Space Telescopes (MAST). Funding for the Kepler mission
is provided by the NASA Science Mission directorate. Wavelet
software was provided by C. Torrence and G. Compo and is
available at http://paos.colorado.edu/research/wavelets/.
REFERENCES
Anfinogentov, S., Nakariakov, V. M., Mathioudakis, M., Van Doorsselaere, T., &
Kowalski, A. F. 2013, ApJ, 773, 156
Balona, L. A., Broomhall, A.-M., Kosovichev, A., et al. 2015, MNRAS,
450, 956
Chen, P. F., & Priest, E. R. 2006, SoPh, 238, 313
Inglis, A. R., & Nakariakov, V. M. 2009, A&A, 493, 259
Kinemuchi, K., Barclay, T., Fanelli, M., et al. 2012, PASP, 124, 963
Kliem, B., Karlický, M., & Benz, A. O. 2000, A&A, 360, 715
Kupriyanova, E. G., Melnikov, V. F., Nakariakov, V. M., & Shibasaki, K.
2010, SoPh, 267, 329
Maehara, H., Shibayama, T., Notsu, S., Notsu, Y., et al. 2012, Natur, 485, 478
Mathioudakis, M., Seiradakis, J. H., Williams, D. R., et al. 2003, A&A,
403, 1101
McLaughlin, J. A., Thurgood, J. O., & MacTaggart, D. 2012, A&A, 548, A98
Mitra-Kraev, U., Harra, L. K., Williams, D. R., & Kraev, E. 2005, A&A,
436, 1041
Murray, M. J., van Driel-Gesztelyi, L., & Baker, D. 2009, A&A, 494, 329
Nakariakov, V. M., Foullon, C., Verwichte, E., & Young, N. P. 2006, A&A,
452, 343
Nakariakov, V. M., Hornsey, C., & Melnikov, V. F. 2012, ApJ, 761, 134
Nakariakov, V. M., Inglis, A. R., Zimovets, I. V., et al. 2010, PPCF, 52,
124009
Nakariakov, V. M., & Melnikov, V. F. 2009, SSRv, 149, 119
Rodono, M. 1974, A&A, 32, 337
Shibata, K., Isobe, H., Hillier, A., et al. 2013, PASJ, 65, 49
Srivastava, A. K., Lalitha, S., & Pandey, J. C. 2013, ApJL, 778, L28
Torrence, C., & Compo, G. P. 1998, BAMS, 79, 61
Van Doorsselaere, T., De Groof, A., Zender, J., Berghmans, D., &
Goossens, M. 2011, ApJ, 740, 90
Van Doorsselaere, T., Nakariakov, V. M., & Verwichte, E. 2007, A&A,
473, 959
Walkowicz, L. M., Basri, G., Batalha, N., et al. 2011, AJ, 141, 50
Welsh, B. Y., Wheatley, J., Browne, S. E., et al. 2006, A&A, 458, 921
Zaitsev, V. V., Kislyakov, A. G., Stepanov, A. V., Kliem, B., & Furst, E. 2004,
AstL, 30, 319
4. SUMMARY
QPPs in the light curve of a flare on KIC 9655129 were
found, which, when analyzed with the wavelet and autocorrelation techniques, showed evidence of the coexistence of
two significant periods of oscillation, namely, 78 ± 12 minutes
and 32 ± 2 minutes. These are consistent with the presence of
two spatial harmonics due to the dispersive nature of guided
MHD waves. Further evidence suggesting that these periodicities are not components of a nonlinear signal was found by
comparing the phase differences and decay times. While it is
possible that these oscillations could be instrumental or
astrophysical artifacts, there is no evidence of periodicities
less than several hours in the rest of the data. Multiple periods
are much more likely to be associated with MHD wave
mechanisms of QPPs, rather than load/unload mechanisms, so
this is one step toward understanding the physics at play and
further suggests that the underlying physics in solar and stellar
flares could be similar. It is also possible that one periodicity is
due to a load/unload mechanism and the other due to an MHD
oscillation.
C.E.P. and V.M.N.: this work was supported by the
European Research Council under the SeismoSun Research
Project No. 321141. V.M.N. acknowledges support from the
STFC consolidated grant ST/L000733/1. A.-M.B. thanks the
Institute of Advanced Study, University of Warwick for their
support. We would like to thank the Kepler team for providing
5